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Edexcel C34 2014 June Q6
10 marks Standard +0.3
6. (a) Express \(\frac { 5 - 4 x } { ( 2 x - 1 ) ( x + 1 ) }\) in partial fractions.
(b) (i) Find a general solution of the differential equation $$( 2 x - 1 ) ( x + 1 ) \frac { \mathrm { d } y } { \mathrm {~d} x } = ( 5 - 4 x ) y , \quad x > \frac { 1 } { 2 }$$ Given that \(y = 4\) when \(x = 2\),
(ii) find the particular solution of this differential equation. Give your answer in the form \(y = \mathrm { f } ( x )\).
Edexcel C34 2014 June Q7
12 marks Moderate -0.3
7. The function f is defined by $$\mathrm { f } : x \mapsto \frac { 3 x - 5 } { x + 1 } , \quad x \in \mathbb { R } , x \neq - 1$$
  1. Find an expression for \(\mathrm { f } ^ { - 1 } ( x )\)
  2. Show that $$\operatorname { ff } ( x ) = \frac { x + a } { x - 1 } , \quad x \in \mathbb { R } , x \neq - 1 , x \neq 1$$ where \(a\) is an integer to be determined. The function \(g\) is defined by $$\mathrm { g } : x \mapsto x ^ { 2 } - 3 x , \quad x \in \mathbb { R } , 0 \leqslant x \leqslant 5$$
  3. Find the value of fg(2)
  4. Find the range of g
Edexcel C34 2014 June Q8
5 marks Standard +0.3
8. The volume \(V\) of a spherical balloon is increasing at a constant rate of \(250 \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\). Find the rate of increase of the radius of the balloon, in \(\mathrm { cm } \mathrm { s } ^ { - 1 }\), at the instant when the volume of the balloon is \(12000 \mathrm {~cm} ^ { 3 }\).
Give your answer to 2 significant figures.
[0pt] [You may assume that the volume \(V\) of a sphere of radius \(r\) is given by the formula \(V = \frac { 4 } { 3 } \pi r ^ { 3 }\).]
Edexcel C34 2014 June Q9
11 marks Standard +0.3
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-13_849_841_214_571} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { e } ^ { \sqrt { x } } , x > 0\) The finite region \(R\), shown shaded in Figure 1, is bounded by the curve, the \(x\)-axis and the lines \(x = 4\) and \(x = 9\)
  1. Use the trapezium rule, with 5 strips of equal width, to obtain an estimate for the area of \(R\), giving your answer to 2 decimal places.
  2. Use the substitution \(u = \sqrt { x }\) to find, by integrating, the exact value for the area of \(R\).
Edexcel C34 2014 June Q10
12 marks Standard +0.8
10. (a) Use the identity for \(\sin ( A + B )\) to prove that $$\sin 2 A \equiv 2 \sin A \cos A$$ (b) Show that $$\frac { \mathrm { d } } { \mathrm {~d} x } \left[ \ln \left( \tan \left( \frac { 1 } { 2 } x \right) \right) \right] = \operatorname { cosec } x$$ A curve \(C\) has the equation $$y = \ln \left( \tan \left( \frac { 1 } { 2 } x \right) \right) - 3 \sin x , \quad 0 < x < \pi$$ (c) Find the \(x\) coordinates of the points on \(C\) where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\) Give your answers to 3 decimal places.
(Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2014 June Q11
12 marks Standard +0.3
11. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-17_600_1024_221_470} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows a sketch of part of the curve \(C\) with equation $$y = \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } , \quad x \in \mathbb { R }$$ where \(a\) is a constant and \(a > \ln 4\) The curve \(C\) has a turning point \(P\) and crosses the \(x\)-axis at the point \(Q\) as shown in Figure 2.
  1. Find, in terms of \(a\), the coordinates of the point \(P\).
  2. Find, in terms of \(a\), the \(x\) coordinate of the point \(Q\).
  3. Sketch the curve with equation $$y = \left| \mathrm { e } ^ { a - 3 x } - 3 \mathrm { e } ^ { - x } \right| , \quad x \in \mathbb { R } , \quad a > \ln 4$$ Show on your sketch the exact coordinates, in terms of \(a\), of the points at which the curve meets or cuts the coordinate axes.
Edexcel C34 2014 June Q12
12 marks Challenging +1.2
12. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-19_568_956_221_502} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve \(C\) with parametric equations $$x = \tan t , \quad y = 2 \sin ^ { 2 } t , \quad 0 \leqslant t < \frac { \pi } { 2 }$$ The finite region \(S\), shown shaded in Figure 3, is bounded by the curve \(C\), the line \(x = \sqrt { 3 }\) and the \(x\)-axis. This shaded region is rotated through \(2 \pi\) radians about the \(x\)-axis to form a solid of revolution.
  1. Show that the volume of the solid of revolution formed is given by $$4 \pi \int _ { 0 } ^ { \frac { \pi } { 3 } } \left( \tan ^ { 2 } t - \sin ^ { 2 } t \right) \mathrm { d } t$$
  2. Hence use integration to find the exact value for this volume.
Edexcel C34 2014 June Q13
11 marks Standard +0.3
13. (a) Express \(2 \sin \theta + \cos \theta\) in the form \(R \sin ( \theta + \alpha )\), where \(R\) and \(\alpha\) are constants, \(R > 0\) and \(0 < \alpha < 90 ^ { \circ }\). Give your value of \(\alpha\) to 2 decimal places.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{423eb549-0873-4185-8faf-12dedafcd108-21_467_1365_870_301} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows the design for a logo that is to be displayed on the side of a large building. The logo consists of three rectangles, \(C , D\) and \(E\), each of which is in contact with two horizontal parallel lines \(l _ { 1 }\) and \(l _ { 2 }\). Rectangle \(D\) touches rectangles \(C\) and \(E\) as shown in Figure 4. Rectangles \(C , D\) and \(E\) each have length 4 m and width 2 m . The acute angle \(\theta\) between the line \(l _ { 2 }\) and the longer edge of each rectangle is shown in Figure 4. Given that \(l _ { 1 }\) and \(l _ { 2 }\) are 4 m apart,
(b) show that $$2 \sin \theta + \cos \theta = 2$$ Given also that \(0 < \theta < 45 ^ { \circ }\),
(c) solve the equation $$2 \sin \theta + \cos \theta = 2$$ giving the value of \(\theta\) to 1 decimal place. Rectangles \(C\) and \(D\) and rectangles \(D\) and \(E\) touch for a distance \(h \mathrm {~m}\) as shown in Figure 4. Using your answer to part (c), or otherwise,
(d) find the value of \(h\), giving your answer to 2 significant figures.
Edexcel C34 2014 June Q14
14 marks Standard +0.8
14. Relative to a fixed origin \(O\), the line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } - 1 \\ - 4 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right)$$ where \(\lambda\) is a scalar parameter. Points \(A\) and \(B\) lie on the line \(l\), where \(A\) has coordinates ( \(1 , a , 5\) ) and \(B\) has coordinates ( \(b , - 1,3\) ).
  1. Find the value of the constant \(a\) and the value of the constant \(b\).
  2. Find the vector \(\overrightarrow { A B }\). The point \(C\) has coordinates ( \(4 , - 3,2\) )
  3. Show that the size of the angle \(C A B\) is \(30 ^ { \circ }\)
  4. Find the exact area of the triangle \(C A B\), giving your answer in the form \(k \sqrt { 3 }\), where \(k\) is a constant to be determined. The point \(D\) lies on the line \(l\) so that the area of the triangle \(C A D\) is twice the area of the triangle \(C A B\).
  5. Find the coordinates of the two possible positions of \(D\).
Edexcel C34 2015 June Q1
9 marks Standard +0.3
  1. A curve has equation
$$4 x ^ { 2 } - y ^ { 2 } + 2 x y + 5 = 0$$ The points \(P\) and \(Q\) lie on the curve.
Given that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2\) at \(P\) and at \(Q\),
  1. use implicit differentiation to show that \(y - 6 x = 0\) at \(P\) and at \(Q\).
  2. Hence find the coordinates of \(P\) and \(Q\).
Edexcel C34 2015 June Q2
9 marks Standard +0.3
2. Given that $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } \equiv \frac { A } { ( 1 - 2 x ) } + \frac { B } { ( 2 + x ) } + \frac { C } { ( 2 + x ) ^ { 2 } }$$
  1. find the values of the constants \(A\) and \(C\) and show that \(B = 0\) (4)
  2. Hence, or otherwise, find the series expansion of $$\frac { 4 \left( x ^ { 2 } + 6 \right) } { ( 1 - 2 x ) ( 2 + x ) ^ { 2 } } , \quad | x | < \frac { 1 } { 2 }$$ in ascending powers of \(x\), up to and including the term in \(x ^ { 2 }\), simplifying each term.
    (5)
Edexcel C34 2015 June Q3
10 marks Standard +0.3
3. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-05_799_885_118_534} \captionsetup{labelformat=empty} \caption{Figure 1}
\end{figure} Figure 1 shows a sketch of part of the curve with equation \(y = \mathrm { f } ( x )\), where $$\mathrm { f } ( x ) = ( 2 x - 5 ) \mathrm { e } ^ { x } , \quad x \in \mathbb { R }$$ The curve has a minimum turning point at \(A\).
  1. Use calculus to find the exact coordinates of \(A\). Given that the equation \(\mathrm { f } ( x ) = k\), where \(k\) is a constant, has exactly two roots,
  2. state the range of possible values of \(k\).
  3. Sketch the curve with equation \(y = | \mathrm { f } ( x ) |\). Indicate clearly on your sketch the coordinates of the points at which the curve crosses or meets the axes.
Edexcel C34 2015 June Q4
7 marks Moderate -0.8
4. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-07_330_494_210_724} \captionsetup{labelformat=empty} \caption{Figure 2}
\end{figure} Figure 2 shows the points \(A\) and \(B\) with position vectors \(\mathbf { a }\) and \(\mathbf { b }\) respectively, relative to a fixed origin \(O\). Given that \(| \mathbf { a } | = 5 , | \mathbf { b } | = 6\) and a.b \(= 20\)
  1. find the cosine of angle \(A O B\),
  2. find the exact length of \(A B\).
  3. Show that the area of triangle \(O A B\) is \(5 \sqrt { 5 }\)
Edexcel C34 2015 June Q5
8 marks Standard +0.3
5. (i) Find the \(x\) coordinate of each point on the curve \(y = \frac { x } { x + 1 } , x \neq - 1\), at which the gradient is \(\frac { 1 } { 4 }\) (ii) Given that $$\int _ { a } ^ { 2 a } \frac { t + 1 } { t } \mathrm {~d} t = \ln 7 \quad a > 0$$ find the exact value of the constant \(a\).
Edexcel C34 2015 June Q6
8 marks Moderate -0.8
6. The mass, \(m\) grams, of a radioactive substance \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { 1 - k t }$$ where \(k\) is a positive constant.
  1. State the value of \(m\) when the radioactive substance was first observed. Given that the mass is 50 grams, 10 years after first being observed,
  2. show that \(k = \frac { 1 } { 10 } \ln \left( \frac { 1 } { 2 } \mathrm { e } \right)\)
  3. Find the value of \(t\) when \(m = 20\), giving your answer to the nearest year.
Edexcel C34 2015 June Q7
8 marks Standard +0.8
7. (a) Use the substitution \(t = \tan x\) to show that the equation $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ can be written in the form $$3 t ^ { 4 } + 8 t ^ { 2 } - 3 = 0$$ (b) Hence solve, for \(0 \leqslant x < 2 \pi\), $$4 \tan 2 x - 3 \cot x \sec ^ { 2 } x = 0$$ Give each answer in terms of \(\pi\). You must make your method clear.
Edexcel C34 2015 June Q8
10 marks Standard +0.3
  1. (a) Prove by differentiation that
$$\frac { \mathrm { d } } { \mathrm {~d} y } ( \ln \tan 2 y ) = \frac { 4 } { \sin 4 y } , \quad 0 < y < \frac { \pi } { 4 }$$ (b) Given that \(y = \frac { \pi } { 6 }\) when \(x = 0\), solve the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \cos x \sin 4 y , \quad 0 < y < \frac { \pi } { 4 }$$ Give your answer in the form \(\tan 2 y = A \mathrm { e } ^ { B \sin x }\), where \(A\) and \(B\) are constants to be determined.
Edexcel C34 2015 June Q9
13 marks Challenging +1.2
9. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-14_709_824_118_559} \captionsetup{labelformat=empty} \caption{Figure 3}
\end{figure} Figure 3 shows a sketch of part of the curve with parametric equations $$x = t ^ { 2 } + 2 t , \quad y = t ^ { 3 } - 9 t , \quad t \in \mathbb { R }$$ The curve cuts the \(x\)-axis at the origin and at the points \(A\) and \(B\) as shown in Figure 3 .
  1. Find the coordinates of point \(A\) and show that point \(B\) has coordinates ( 15,0 ).
  2. Show that the equation of the tangent to the curve at \(B\) is \(9 x - 4 y - 135 = 0\) The tangent to the curve at \(B\) cuts the curve again at the point \(X\).
  3. Find the coordinates of \(X\).
    (Solutions based entirely on graphical or numerical methods are not acceptable.)
Edexcel C34 2015 June Q10
6 marks Standard +0.3
10. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-16_319_508_237_719} \captionsetup{labelformat=empty} \caption{Figure 4}
\end{figure} Figure 4 shows a right circular cylindrical rod which is expanding as it is heated.
At time \(t\) seconds the radius of the rod is \(x \mathrm {~cm}\) and the length of the rod is \(6 x \mathrm {~cm}\).
Given that the cross-sectional area of the rod is increasing at a constant rate of \(\frac { \pi } { 20 } \mathrm {~cm} ^ { 2 } \mathrm {~s} ^ { - 1 }\), find the rate of increase of the volume of the rod when \(x = 2\) Write your answer in the form \(k \pi \mathrm {~cm} ^ { 3 } \mathrm {~s} ^ { - 1 }\) where \(k\) is a rational number.
Edexcel C34 2015 June Q11
13 marks Standard +0.3
11. (a) Express \(1.5 \sin \theta - 1.2 \cos \theta\) in the form \(R \sin ( \theta - \alpha )\), where \(R > 0\) and \(0 < \alpha < \frac { \pi } { 2 }\) Give the value of \(R\) and the value of \(\alpha\) to 3 decimal places. The height, \(H\) metres, of sea water at the entrance to a harbour on a particular day, is modelled by the equation $$H = 3 + 1.5 \sin \left( \frac { \pi t } { 6 } \right) - 1.2 \cos \left( \frac { \pi t } { 6 } \right) , \quad 0 \leqslant t < 12$$ where \(t\) is the number of hours after midday.
(b) Using your answer to part (a), calculate the minimum value of \(H\) predicted by this model and the value of \(t\), to 2 decimal places, when this minimum occurs.
(c) Find, to the nearest minute, the times when the height of sea water at the entrance to the harbour is predicted by this model to be 4 metres.
Edexcel C34 2015 June Q12
10 marks Standard +0.3
  1. (i) Relative to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation
$$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } - 5 \\ 1 \\ 6 \end{array} \right) + \lambda \left( \begin{array} { r } 2 \\ - 3 \\ 1 \end{array} \right) \text { where } \lambda \text { is a scalar parameter. }$$ The point \(P\) lies on \(l _ { 1 }\). Given that \(\overrightarrow { O P }\) is perpendicular to \(l _ { 1 }\), calculate the coordinates of \(P\).
(ii) Relative to a fixed origin \(O\), the line \(l _ { 2 }\) is given by the equation $$l _ { 2 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ - 3 \\ 12 \end{array} \right) + \mu \left( \begin{array} { r } 5 \\ - 3 \\ 4 \end{array} \right) \text { where } \mu \text { is a scalar parameter. }$$ The point \(A\) does not lie on \(l _ { 2 }\). Given that the vector \(\overrightarrow { O A }\) is parallel to the line \(l _ { 2 }\) and \(| \overrightarrow { O A } | = \sqrt { 2 }\) units, calculate the possible position vectors of the point \(A\).
Edexcel C34 2015 June Q13
14 marks Standard +0.3
13. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{4c08fbab-283e-4c92-89a4-10f68f37e133-22_536_929_223_504} \captionsetup{labelformat=empty} \caption{Figure 5}
\end{figure} Figure 5 shows a sketch of part of the curve with equation \(y = 2 - \ln x , x > 0\) The finite region \(R\), shown shaded in Figure 5, is bounded by the curve, the \(x\)-axis and the line with equation \(x = \mathrm { e }\). The table below shows corresponding values of \(x\) and \(y\) for \(y = 2 - \ln x\)
\(x\)e\(\frac { \mathrm { e } + \mathrm { e } ^ { 2 } } { 2 }\)\(\mathrm { e } ^ { 2 }\)
\(y\)10
  1. Complete the table giving the value of \(y\) to 4 decimal places.
  2. Use the trapezium rule, with all the values of \(y\) in the completed table, to obtain an estimate for the area of \(R\), giving your answer to 3 decimal places.
  3. Use integration by parts to show that \(\int ( \ln x ) ^ { 2 } \mathrm {~d} x = x ( \ln x ) ^ { 2 } - 2 x \ln x + 2 x + c\) The area \(R\) is rotated through \(360 ^ { \circ }\) about the \(x\)-axis.
  4. Use calculus to find the exact volume of the solid generated. Write your answer in the form \(\pi \mathrm { e } ( p \mathrm { e } + q )\), where \(p\) and \(q\) are integers to be found.
Edexcel C34 2017 June Q1
6 marks Moderate -0.3
  1. A curve \(C\) has equation
$$3 x ^ { 2 } + 2 x y - 2 y ^ { 2 } + 4 = 0$$ Find an equation for the tangent to \(C\) at the point ( 2,4 ), giving your answer in the form \(a x + b y + c = 0\) where \(a , b\) and \(c\) are integers.
(6)
Edexcel C34 2017 June Q2
6 marks Moderate -0.3
  1. Use integration by parts to find the exact value of \(\int _ { 1 } ^ { \mathrm { e } } \frac { \ln x } { x ^ { 2 } } \mathrm {~d} x\)
Write your answer in the form \(a + \frac { b } { \mathrm { e } }\), where \(a\) and \(b\) are integers.
Edexcel C34 2017 June Q3
8 marks Moderate -0.3
3. The function g is defined by $$g ( x ) = \frac { 6 x } { 2 x + 3 } \quad x > 0$$
  1. Find the range of g .
  2. Find \(\mathrm { g } ^ { - 1 } ( x )\) and state its domain.
  3. Find the function \(\operatorname { gg } ( x )\), writing your answer as a single fraction in its simplest form.